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Rarity, significance, surprise

Take two shuffled, standard 52-card decks. Draw a card from each. If you find them to be the same card, that’s rather surprising, no? But what if you find the first card to be the jack of hearts, and the second card the four of clubs? That’s not very surprising, even though it’s fifty-two times less likely to occur than getting two of the same card!

I think part of the answer as to why the pair of identical cards is more surprising than the much rarer Jh4c draw is that we’re implicitly entertaining an alternative hypothesis. The null hypothesis of course is that both decks are completely fair and uncorrelated — nothing fishy’s going on. The alternative is that something fishy is indeed going on. And our notions of fishiness get triggered more by a recognizable pattern than by something that doesn’t fit into any pattern. Conceivably, we could have a different sense of patterns. Just as a draw of something like 9s9s is maximally concordant, a draw of Jh4c is maximally discordant. Jacks and fours are as far from each other as they can be in number, (assuming circularity where A lies between K and 2), and also in suit: the standard suit order is clubs, diamonds, hearts, spades, so, with similar circularity assumptions, clubs and hearts are antipodal. If you don’t fully buy this, well, they’re different colors in any event.

You have to take my word on this, but when I picked the jack of hearts and the four of clubs, I tried to think of the most random-sounding pair of cards I could, and I ended up with picking a maximally discordant pair, which is just as improbable as picking a maximally concordant pair (i.e., two of the same card). That’s pretty surprising, no? But only now is it surprising, since your sense of patterns has been updated to include this concordance metric as a means by which to evaluate draws.

A test that’s purely a test of the null hypothesis is impossible. You’re always comparing two hypothesis. It’s better to understand what both of them are than to only understand one.